Abstract

We provide a novel perspective on “regularity” as a property of representations of the Weyl algebra. We first critique a proposal by Halvorson [2004, “Complementarity of representations in quantum mechanics”, Studies in History and Philosophy of Modern Physics35 (1), pp. 45–56], who argues that the non-regular “position” and “momentum” representations of the Weyl algebra demonstrate that a quantum mechanical particle can have definite values for position or momentum, contrary to a widespread view. We show that there are obstacles to such an intepretation of non-regular representations. In Part II, we propose a justification for focusing on regular representations, pace Halvorson, by drawing on algebraic methods.

Source: Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics

Author(s): Benjamin Feintzeig, James Owen Weatherall

Abstract

We provide a novel perspective on “regularity” as a property of representations of the Weyl algebra. In Part I, we critiqued a proposal by Halvorson [2004, “Complementarity of representations in quantum mechanics”, Studies in History and Philosophy of Modern Physics35 (1), pp. 45–56], who advocates for the use of the non-regular “position” and “momentum” representations of the Weyl algebra. Halvorson argues that the existence of these non-regular representations demonstrates that a quantum mechanical particle can have definite values for position or momentum, contrary to a widespread view. In this sequel, we propose a justification for focusing on regular representations, pace Halvorson, by drawing on algebraic methods.

Along the years, supersymmetric quantum mechanics (SUSY QM) has been used for studying solvable quantum potentials. It is the simplest method to build Hamiltonians with prescribed spectra in the spectral design. The key is to pair two Hamiltonians through a finite order differential operator. Some related subjects can be simply analyzed, as the algebras ruling both Hamiltonians and the associated coherent states. The technique has been applied also to periodic potentials, where the spectra consist of allowed and forbidden energy bands. In addition, a link with non-linear second order differential equations, and the possibility of generating some solutions, can be explored. Recent applications concern the study of Dirac electrons in graphene placed either in electric or magnetic fields, and the analysis of optical systems whose relevant equations are the same as those of SUSY QM. These issues will be reviewed briefly in this paper, trying to identify the most important subjects explored currently in the literature.

The Schrodinger and Heisenberg pictures are equivalent formulations of quantum mechanics in the sense that they give the same expectation value for any operator. We consider a sequence of two or more unitary transformations and show that the Heisenberg operator produced after the first transformation cannot be viewed as the input to the second transformation. The experimental consequences of this are illustrated by several examples in quantum optics.

Although the wish to unify theories into something more fundamental is omnipresent and compelling, nonetheless, in a sense, theories must first be unifiable. The reasons for the success of the unification of electricity and magnetism into a theory of electromagnetism are contrasted with the reasons for the failure of the Einstein-Maxwell unification of gravitation and electromagnetism and the attempts of quantum gravity to unify Einstein’s theory of gravity with quantum field theory. The difference between a unification of two theories, a concatenation of them, and the existence of a formal analogy between them is also discussed.

Source: Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics

Author(s): Michael E. Cuffaro

Abstract

The principle of ‘information causality’ can be used to derive an upper bound—known as the ‘Tsirelson bound’—on the strength of quantum mechanical correlations, and has been conjectured to be a foundational principle of nature. To date, however, it has not been sufficiently motivated to play such a foundational role. The motivations that have so far been given are, as I argue, either unsatisfactorily vague or appeal to little if anything more than intuition. Thus in this paper I consider whether some way might be found to successfully motivate the principle. And I propose that a compelling way of so doing is to understand it as a generalisation of Einstein’s principle of the mutually independent existence—the ‘being-thus’—of spatially distant things. In particular I first describe an argument, due to Demopoulos, to the effect that the so-called ‘no-signalling’ condition can be viewed as a generalisation of Einstein’s principle that is appropriate for an irreducibly statistical theory such as quantum mechanics. I then argue that a compelling way to motivate information causality is to in turn consider it as a further generalisation of the Einsteinian principle that is appropriate for a theory of communication. I describe, however, some important conceptual obstacles that must yet be overcome if the project of establishing information causality as a foundational principle of nature is to succeed.

Source: Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics

Author(s): Baptiste Le Bihan, Niels Linnemann

Abstract

Important features of space and time are taken to be missing in quantum gravity, allegedly requiring an explanation of the emergence of spacetime from non-spatio-temporal theories. In this paper, we argue that the explanatory gap between general relativity and non-spatio-temporal quantum gravity theories might significantly be reduced with two moves. First, we point out that spacetime is already partially missing in the context of general relativity when understood from a dynamical perspective. Second, we argue that most approaches to quantum gravity already start with an in-built distinction between structures to which the asymmetry between space and time can be traced back.

Gradient-based hybrid quantum-classical algorithms are often initialised with random, unstructured guesses. Here, the authors show that this approach will fail in the long run, due to the exponentially-small probability of finding a large enough gradient along any direction.

How does gravity work at the particle level? The question has stumped physicists since the two bedrock theories of general relativity (Albert Einstein’s equations envisioning gravity as curves in the geometry of space-time) and quantum mechanics (equations that describe particle interactions) revolutionized the discipline about a century ago.

One challenge to solving the problem lies in the relative weakness of gravity compared with the strong, weak and electromagnetic forces that govern the subatomic realm. Though gravity exerts an unmistakable influence on macroscopic objects like orbiting planets, leaping sharks and everything else we physically experience, it produces a negligible effect at the particle level, so physicists can’t test or study how it works at that scale.

Confounding matters, the two sets of equations don’t play well together. General relativity paints a continuous picture of space-time while in quantum mechanics everything is quantized in discrete chunks. Their incompatibility leads physicists to suspect that a more fundamental theory is needed to unify all four forces of nature and describe them at all scales.

One relatively recent approach to understanding quantum gravity makes use of a “holographic duality” from string theory called the AdS-CFT correspondence. Our latest In Theory video explains how this correspondence connects a lower dimensional particle theory to a higher dimensional space that includes gravity:

This holographic duality has become a powerful theoretical tool in the quest to understand quantum gravity and the inner workings of black holes and the Big Bang, where extreme gravity operates at tiny scales.

We hope you enjoyed this second episode from season two of Quanta’s In Theory video series. Season two opened in August with an animated explainer about a mysterious mathematical pattern that has been discovered in disparate settings — in the energy spectra of heavy atomic nuclei, a function related to the distribution of prime numbers, an independent bus system in Mexico, spectral measurements of the internet, Arctic ponds, human bones and the color-sensitive cone cells in chicken eyes. To learn more, watch episode one below:

The traditional formalism of nonrelativistic quantum theory allows the state of a quantum system to extend across space, but only restricts it to a single instant in time, leading to distinction between theoretical treatments of spatial and temporal quantum correlations. Here we unify the geometrica…

The local conservation of a physical quantity whose distribution changes with time is mathematically described by the continuity equation. The corresponding time parameter, however, is defined with respect to an idealized classical clock. We consider what happens when this classical time is replaced…

Electrons are confined to an artificial Sierpiński triangle. Microscopy measurements show that their wavefunctions become self-similar and their quantum properties inherit a non-integer dimension between 1 and 2.